Berkant Tacer

ON THE AMPLITUDE- AND FREQUENCY-MODULATION DECOMPOSITION OF SIGNALS

In general, the problem of determining the amplitude and frequency modulations (AM and FM) of a signal is ill posed because there is an unlimited number of combinations of AM and FM that will generate a given signal. Although Gabor proposed a method for uniquely defining the AM and FM of a signal, namely via the analytic signal, the results obtained are sometimes physically paradoxical. In this paper, four reasonable physical conditions that the calculated AM and FM of a signal should satisfy are proposed. The analytic signal method generally fails to satisfy two of the four conditions. A method utilizing the positive (Cohen–Posch) time‐frequency distribution and time‐varying coherent demodulation of the signal is given for obtaining an AM and FM that satisfy the four proposed conditions. Contrary to the accepted definition, the instantaneous frequency (i.e., the FM) that satisfies these conditions is generally not the derivative of the phase of the signal. Rather, the phase is separated into two parts, o...

Comments on the interpretation of instantaneous frequency

Instantaneous frequency, taken as the derivative of the phase of the signal, is interpreted in the time-frequency literature as the average frequency of the signal at each time. We point out some difficulties with this interpretation, and show that for a generic two-component AM-FM signal, the interpretation holds only when the components are of equal strength. We conclude that instantaneous frequency and the average frequency at each time are generally two different quantities. One possible interpretation of the difference between these two quantities is suggested.

NON-STATIONARY SIGNAL CLASSIFICATION USING THE JOINT MOMENTS OF TIME-FREQUENCY DISTRIBUTIONS

We present a time-frequency based non-stationary time-series classification method which utilizes features derived from the joint moments of time-frequency distributions (TFDs). The method is applied to both synthetic and real signals, with comparison to classification performance utilizing features derived from temporal moments only and spectral moments only. The results show that a classification algorithm which utilizes joint time-frequency information, as quantified by the joint moments of the TFD, can improve performance over time or frequency-based features alone, for classification of non-stationary time series.

Instantaneous frequency and the conditional mean frequency of a signal

Abstract Time-varying frequency is a natural occurrence, the mathematical and physical description of which has been evolving for many decades. One description of time-varying frequency is the instantaneous frequency proposed by Gabor, defined as the derivative of the phase of the (analytic) signal. The interpretation of this quantity has been a subject of much investigation. One interpretation arising from time—frequency distribution theory is that instantaneous frequency is the average frequency at each time in the signal. We explore this interpretation in detail, and derive conditions on an arbitrary two-component AM—FM complex signal for which this interpretation is plausible. The situations for which these conditions are met are limited. We also show that while one can force the interpretation by generating a new complex representation (not necessarily analytic) for which the derivative of the phase coincides with the average frequency at each time and for which the real part of this complex signal is the given signal, the amplitude of this complex signal is generally unbounded. Thus, if instantaneous frequency is to interpreted as the average frequency at each time, the instantaneous amplitude must generally be unbounded. Conversely, if we insist that the instantaneous amplitude be bounded, then the instantaneous frequency generally cannot be interpreted as the average frequency at each time.

What are the joint time-frequency moments of a signal?

We investigate the question of what the joint moments of a signal are by considering the joint moments of the spectrogram for limiting cases of the window. Operator methods are also explored. Expressions for the joint moments are derived, which reveal the distorting effects of the spectrogram window. Knowledge of the joint moments of a signal may be useful in estimating positive time-frequency distributions, or in signal classification of nonstationary signals.

Time-frequency-based classification

We propose a time-frequency based pattern classification method which utilizes the joint moments of time-frequency distributions (TFDs) for features. The method is applied to a biomedical data set, and compared to a template matching scheme and to methods utilizing only temporal moments or spectral moments. Our results show that a classification algorithm which utilizes joint time-frequency information, as quantified by the joint moments of the TFD, can potentially improve performance over time or frequency-based methods alone, for classification of nonstationary time series.

Time-scale energy density functions

Scale, like frequency, is a physical characteristic of a signal. To measure the scale content of a signal, the signal must be appropriately transformed. A theory for joint time-scale energy density functions is presented, and a method for generating such functions for any signal is given. Examples for synthetic signals and real data are presented. The theory and method can be extended to arbitrary joint densities of any variables, for example, frequency and scale.

Maximum entropy time‐frequency analysis

A maximum entropy method for constructing a joint time‐frequency density of a signal is presented. The method constructs the density from a set of joint time‐frequency moments that are functions of the signal and spectral amplitudes and phases. As such, the moments can be obtained a priori from the signal, independent of the density. Unlike the Wigner and other quadratic time‐frequency distributions, the maximum entropy density is non‐negative and unhindered by cross terms. Also, unlike the spectrogram, it is not limited by a resolution trade‐off between time and frequency. Examples are given to demonstrate the method, including maximum entropy densities of speech, bat sonar and acoustic scattering signals. [Supported by ONR.]

Positive time-scale distributions

We define the class of positive time-scale distributions (TSDs), and give a general method for constructing such functions. We explore various properties and moments of these distributions. Examples, of both synthetic and real signals, are given.<<ETX>>

A time-frequency training-based approach for robust classification of unknown transients with unknown arrival time and doppler shift

Abstract We present a training-based approach for the classification of noisy unknown transient signals with arbitrary range and Doppler shift (time and frequency shifts). The ambiguity function, which is the 2-D inverse Fourier transform of the Wigner time-frequency distribution of the signal, is utilized to remove the unknown time and frequency shifts. An ambiguity domain template is then generated from labeled training data (tens of observations), and classification is performed using an inner product. The method is tested on synthetic transient signals in Gaussian noise and performs as well as or better than another recently proposed time-frequency based method, and an energy detector, particularly when limited training data are available.